\title{Resolutions over symmetric algebras with radical cube zero}
\author{David J. Benson}
\address{Department of Mathematics, University of Aberdeen,
Aberdeen AB24 3UE}

\begin{abstract}
Let $\Lambda$ be a finite dimensional indecomposable symmetric
algebra over an algebraically closed field $k$,
satisfying $J^3(\Lambda)=0$. Let $S_1,\dots,S_r$ 
be representatives of the isomorphism classes of simple $\Lambda$-modules,
and let $E$ be the $r\times r$ matrix whose $(i,j)$ entry is 
$\dim_k \Ext^1_\Lambda(S_i,S_j)$. 
If there exists an eigenvalue $\lambda$
of $E$ satisfying $|\lambda|>2$ then the minimal resolution of each 
non-projective finitely generated 
$\Lambda$-module has exponential growth, with radius
of convergence $\frac{1}{2}(\lambda - \sqrt{\lambda^2 - 4})$.
On the other hand, 
if all eigenvalues $\lambda$ of $E$ 
satisfy $|\lambda|\leq 2$ then the dimensions of the modules in the 
minimal projective resolution of
each finitely generated $\Lambda$-module are either bounded or grow linearly.
In this case, we classify the possibilities for the matrix $E$.
The proof is an application of the Perron--Frobenius theorem. 
\end{abstract}